When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.

翻译：当通过使用专门的近似技术来接近椭圆性问题时,我们获得大型结构矩阵,其分析可提供关于方法稳定性的信息;我们在这里通过临时的有限差异,为拉普拉西亚近似的矩阵序列提供光谱和标准估计;分析涉及矩阵理论的若干工具,特别是托普利茨操作员和通用本地托普利茨矩阵序列的设置;进行了若干次数字实验,证实了理论结论的正确性。